**Section formula :**

We use the section formula to find the point which divides the line segment in a given ratio.

The point P which divides the line segment joining the two points A (x_{1}, y_{1}) and B (x_{2}, y_{2}) internally in the ratio l : m is

If P divides a line segment AB joining the two points A (x_{1}, y_{1}) and B (x_{2}, y_{2}) externally in the ratio l : m is,

Let us look into some example problems based on the above concepts.

**Example 1 :**

Find the point which divide the line segment joining the points (-1,2) and (4,-5) internally in the ratio 2 : 3.

**Solution :**

Since we need to find the point which divides the line segment internally in the ratio 2 : 3, we have to use the formula

= (lx_{2} + mx_{1}) / (l + m), (ly_{2} + my_{1}) / (l + m)

Here (x_{1}, y_{1}) ==> (-1, 2) (x_{2}, y_{2}) ==> (4, -5) and l : m ==> 2:3

= 2(4) + 3(-1) / (2 + 3) , 2(-5) + 3(2) / (2 + 3)

= [(8 - 3) / 5 , (-10 + 6) / 5]

= (5 / 5 , -4 / 5)

= (1, -4/5)

Hence the point (1, -4/5) divides the line segment internally in the ratio 2 : 3.

**Example 2 :**

Find the point which divide the line segment joining the points (2, 1) and (3, 5) externally in the ratio 2 : 3.

**Solution :**

Since we need to find the point which divides the line segment externally in the ratio 2 : 3, we have to use the formula

= (lx_{2} - mx_{1}) / (l - m), (ly_{2} - my_{1}) / (l - m)

Here (x_{1}, y_{1}) ==> (2, 1) (x_{2}, y_{2}) ==> (3, 5) and l : m ==> 2:3

= 2(3) - 3(2) / (2 - 3) , 2(5) - 3(1) / (2 - 3)

= [(6 - 6) / 5 , (10 - 3) / 5]

= (0 / 5 , 7 / 5)

= (0, 7/5)

Hence the point (0, 7/5) divides the line segment externally in the ratio 2 : 3.

**Example 3 :**

Find the ratio in which x axis divides the line segment joining the points (6 , 4) and (1 ,- 7).

**Solution :**

Let l : m be the ratio of the line segment joining the points (6,4) and (1,-7) and let p(x,0) be the point on the x axis

Section formula internally

= (lx₂ + mx₁)/(l + m) , (ly₂ + my₁)/(l + m)

(x, 0) = [(1) + m(6)]/(l + m) , [l(-7) + m(4)]/(l + m)

(x , 0) = [l + 6 m]/(l + m) , [-7l + 4m]/(l + m)

Equating y-coordinates

[-7l + 4m]/(l + m) = 0

- 7 l + 4 m = 0

- 7 l = - 4 m

l/m = 4/7

l : m = 4 : 7

Hence x-axis divides the line segment in the ratio 4 : 7.

After having gone through the stuff given above, we hope that the students would have understood "Section Formula ".

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